Research Proposal on Arithmetic Geometry

柴教授将继续研究主极化阿贝尔变体模空间中线性子变体的性质。他还将考虑精细结构的还原志村品种的pel型，。这是他最近关于pel型的Shimura变种的普通Hecke轨道密度的结果的自然延伸。在一个新的研究方向上，柴教授将尝试将l进上同调技术应用于最近图论工作中出现的特定指数和。这项工作在数论的一个现代领域被称为算术代数几何。数论是研究计数数，1、2、3等的学科，是数学中最古老的分支。正如它复杂的名字所暗示的那样，算术代数几何是数学三个不同领域的综合。在过去的50年里，数学家们发现，从几何学中熟悉的物体，比如多项式方程图中的曲线，可以通过将它们视为抽象的代数结构来更好地理解。这个强大的思想彻底改变了几何学。随着时间的推移，数论学家意识到，他们的许多最困难的问题都可以用这些抽象的几何对象的代数描述来提出和解决。正如数学中经常发生的那样，这种不断提高的抽象水平实际上产生了许多非常具体和实用的结果。

Ching-Li Chai9800609Professor Chai will continue his study of the properties of linear subvarieties in the moduli space of principally polarized abelian varieties. He will also consider the fine structure of the reduction of the Shimura varieties of PEL-type,. This is a natural extension of his recent results on the density of ordinary Hecke orbits of Shimura varieties of PEL-type. In a new direction of investigation, Professor Chai will try to apply the technique of l-adic cohomology to particular exponential sums that have arisen in recent work in graph theory.This work in a modern area of Number Theory known as arithmetic algebraic geometry. Number Theory is the study of the counting numbers, 1, 2, 3, etc. and is the oldest branch of mathematics. Just as the complicated name suggests, arithmetic algebraic geometry is a synthesis of three different areas of mathematics. In the last fifty years, mathematicians have found that familiar objects from geometry, like the curves that appear as the graph of a polynomial equation, can be much better understood by viewing them as abstract algebraic constructions. This powerful idea revolutionized geometry. In time number theorists realized that many of their most difficult problems could be posed and then solved in terms of these abstract algebraic descriptions of geometric objects. As often happens in mathematics, this ever increasing level of abstraction has actually produced many very concrete and practical results.